nLab real reciprocal function

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Analysis

Algebra

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Definition

Implicit definition

In real analysis, the reciprocal 1x\frac{1}{x} is a partial function implicitly defined over the non-zero real numbers by the equation x(1x)=1x \left(\frac{1}{x}\right) = 1. This is the definition commonly used when defining the real numbers as a field.

By the exponential and natural logarithm functions

The reciprocal 1():(,0)(0,)\frac{1}{(-)}:(-\infty,0)\union(0,\infty)\to\mathbb{R} is piecewise defined as

1x{e ln(x) x(,0) e ln(x) x(0,) \frac{1}{x} \coloneqq \begin{cases} -e^{-\ln(-x)} & x \in (-\infty,0) \\ e^{-\ln(x)} & x \in (0,\infty) \end{cases}

This definition implies that the reciprocal is analytic in each of the two connected components of the domain.

By power series

Let us define the functions f:(1,1)f \colon (-1,1)\to\mathbb{R} and g:(1,1)g:(-1,1)\to\mathbb{R} from the open subinterval of the real numbers (1,1)(-1,1) \subset \mathbb{R} to the real numbers \mathbb{R} as the locally convergent power series

f(x) n=0 x n f(x)\coloneqq \sum_{n=0}^{\infty} x^n
g(x) n=0 (1) nx n g(x)\coloneqq \sum_{n=0}^{\infty} (-1)^n x^n

The reciprocal 1():(,0)(0,)\frac{1}{(-)}:(-\infty,0)\union(0,\infty)\to\mathbb{R} is then piecewise defined as

1x{lim a0 (a) n=0 a n(xf(a+1)) n x(,0) lim a0 +a n=0 (a) n(x+g(a1)) n x(0,) \frac{1}{x} \coloneqq \begin{cases} \lim_{a\to 0^-} (-a) \sum_{n=0}^{\infty} a^n (x-f(a+1))^n & x \in (-\infty,0) \\ \lim_{a\to 0^+} a \sum_{n=0}^{\infty} (-a)^n (x+g(a-1))^n & x \in (0,\infty) \end{cases}

Equivalently, given an element p(0,1)p \in (0, 1), the reciprocal 1():(,0)(0,)\frac{1}{(-)}:(-\infty,0)\union(0,\infty)\to\mathbb{R} is then piecewise defined as

1x{lim ip i n=0 (p i) n(xf(p i+1)) n x(,0) lim ip i n=0 (p i) n(x+g(p i1)) n x(0,) \frac{1}{x} \coloneqq \begin{cases} \lim_{i \to \infty} p^i \sum_{n=0}^{\infty} (-p^i)^n (x-f(-p^i+1))^n & x \in (-\infty,0) \\ \lim_{i \to \infty} p^i \sum_{n=0}^{\infty} (-p^i)^n (x+g(p^i-1))^n & x \in (0,\infty) \end{cases}

This definition implies that the reciprocal is analytic in each of the two connected components of the domain.

Last revised on August 21, 2024 at 01:54:54. See the history of this page for a list of all contributions to it.